Non-Linear Modal Interactions in Shallow Suspended Cables

Pilipchuk, Valery N.; Ibrahim, R. A.

doi:10.1006/jsvi.1999.2326

Cited by 21 publications

(6 citation statements)

References 19 publications

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“…A special type of coordinate transformation which provides a description of the in-plane and out-of-plane motions was proposed for shallow shells [87] and shallow suspended cables [88]. The new coordinates are constructed along and are perpendicular to the unstretched manifold.…”

Section: Other Approachesmentioning

confidence: 99%

Asymptotic construction of nonlinear normal modes for continuous systems

Andrianov

2007

Nonlinear Dyn

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This paper provides a review of some asymptotic methods for construction of nonlinear normal modes for continuous system (NNMCS). Asymptotic methods of solving problems relating to NNMCS have been developed by many authors. The main features of this paper are that (i) it is devoted to the basic principles of asymptotic approaches for constructing of NNMCS; (ii) it deals with both traditional approaches and, less widely used, new approaches; and (iii) it pays a lot of attention to the analysis of widely used simplified mechanical models for the analysis of NNMCS. The author has paid special attention to examples and discussion of results. Keywords Nonlinear oscillations . Normal mode . Asymptotic approach . Continuous system 1 Definition of nonlinear normal modes for continuous system First definition of nonlinear normal modes for continuous system (NNMCS) is based on exact separation of spatial and time variables [1]. Exact separation of spatial and time variables is possible for some simplified models of continuous systems (Kirchhoff model of nonlinear beam [2], Berger model of nonlinear plate [3,4], Berger-like model of shallow shell [5-7]) for simply supported ends. Naturally, solutions of this type were constructed earlier [2], but Wah was the first who pointed to links between these solutions and those proposed by R.M. Rosenberg for discrete system concepts of NNM [8][9][10]. Naturally, area of applicability of Wah definition is very restricted.Shaw and Pierre proposed the following definition of NNMCS [11]: "If the displacement u 0 (t) = u(s 0 , t) and velocity v 0 (t) = v(s 0 , t) of a single point, s = s 0 , are known, then the entire displacement and velocity field can be determined by the dynamics of that single point u(s, t) = U (u 0 (t), v 0 (t), s, s 0 ), v(s, t) = V (u 0 (t), v 0 (t), s, s 0 )."

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Section: Other Approachesmentioning

confidence: 99%

Asymptotic construction of nonlinear normal modes for continuous systems

Andrianov

2007

Nonlinear Dyn

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“…Rega, Lacarbonara, Nayfeh, and Chin [12] considered multimodal resonances in a suspended cable and studied them using the perturbation approach. Pilipchuk and Ibrahim also investigated nonlinear mode interactions for a horizontal cable [13]. Zheng, Ko, and Ni [14] considered the super-harmonics and internal resonance of a suspended cable with almost commensurable natural frequencies.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations and chaos of an inclined cable

Chen

2008

Nonlinear Dyn

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The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin's method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of perioddoubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kovačič and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the

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“…In the past decades, there has been considerable interest in the study of modal interactions of suspended cables subjected to harmonic excitations at primary resonances (Benedettini et al, 1986;Rao and Iyengar, 1991;Perkins, 1992;Perkins, 1993, 1995;Benedettini et al, 1995;Pakdemirli et al, 1995;Rega et al, 1999;Nayfeh et al, 2002;Zhang and Tang, 2002;Gattulli et al, 2004), at random excitations (Chang et al, 1996;Chang and Ibrahim, 1997;Ibrahim and Chang, 1999), and at complex excitations including wind excitations (Takahashi and Wang, 1990;Luongo and Piccardo, 1998;Martinelli and Perotti, 2001), fluid excitations (Kim and Perkins, 2002), by using the multi-degree-of-freedom (MDOF) models. Furthermore, Pilipchuk and Ibrahim (2002) introduced a special type of coordinate transformation to examine different regimes of nonlinear modal interactions of suspended cables.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Wang

Zhao

2006

International Journal of Solids and Structures

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Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton-Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force-response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.

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Non-Linear Modal Interactions in Shallow Suspended Cables

Cited by 21 publications

References 19 publications

Asymptotic construction of nonlinear normal modes for continuous systems

Asymptotic construction of nonlinear normal modes for continuous systems

Bifurcations and chaos of an inclined cable

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

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